We study the ergodic properties of a class of measures on $\unicode[STIX]{x1D6F4}^{\mathbb{Z}}$ for which $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\Vert A_{x_{0}}\cdots A_{x_{n-1}}\Vert ^{t}$, where ${\mathcal{A}}=(A_{0},\ldots ,A_{M-1})$ is a collection of matrices. The measure $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$ is called a matrix Gibbs state. In particular, we give a sufficient condition for a matrix Gibbs state to have the weak Bernoulli property. We employ a number of techniques to understand these measures, including a novel approach based on Perron–Frobenius theory. We find that when $t$ is an even integer the ergodic properties of $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$ are readily deduced from finite-dimensional Perron–Frobenius theory. We then consider an extension of this method to $t>0$ using operators on an infinite- dimensional space. Finally, we use a general result of Bradley to prove the main theorem.
In this paper we give a short proof that the projection of a Gibbs state for a Hölder continuous potential on a mixing shift of finite type under a 1block fiber-wise mixing factor map has a Hölder continuous g function. This improves a number of previous results. The key insight in the proof is to realize the measure of a cylinder set in terms of positive operators and use cone techniques.
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